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A243770 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).

%I A243770 #10 Jun 11 2014 19:56:22
%S A243770 1,1,3,11,16,57,161,927,1997,5539,25638,68850,275765,995088,2784600,
%T A243770 19235059,53549250,177389053,711629836,2641203240,7517769634,
%U A243770 31706388438,147201204924,455738363552,1614252170849,6020919907344,23811404216400,79787485940824
%N A243770 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).
%H A243770 Alois P. Heinz, <a href="/A243770/b243770.txt">Table of n, a(n) for n = 1..400</a>
%e A243770 a(1) = 1: (U)D.
%e A243770 a(2) = 1: U(UD)D.
%e A243770 a(3) = 3: UD(UU)DD, (UU)DDUD, (UU)DUDD.
%e A243770 a(4) = 11: UDUDU(UDD), UDU(UDD)UD, UDUUD(UDD), UDUU(UDD)D, U(UDD)UDUD, UUD(UDD)UD, UUDUD(UDD), UUDU(UDD)D, UU(UDD)DUD, UUUD(UDD)D, UUU(UDD)DD.
%Y A243770 Column k=1 of A243752.
%Y A243770 Main diagonal of A243827.
%K A243770 nonn
%O A243770 1,3
%A A243770 _Alois P. Heinz_, Jun 10 2014